Limits and Continuity

1. Short lesson on the definition of the derivative

link to lesson


2. Lesson on the definition of the derivative (it is a bit long; you may wish to see only 13:15 to 20:40)





3. Review lesson on slopes of tangents and instantaneous rate of change (from MHF4U)





An example requiring 'Rationalizing the Numerator







4. Calculating Areas under a Curve (calculating an 'integral')

(i) Detailed calculations of "Upper" rectangles, "Lower" rectangles and rectangles formed from midpoints of function



(ii) Another Example



(iii) Still Another Example (ignore the fancy integral notation he introduces at the beginning; it isn't necessary)




The Limit of a Function

1. Introduction and definition



Another Overview



2. Evaluate limits by simplifying the expression or by factoring



3. Evaluate a limit by rationalizing the numerator / denominator




4. Evaluate a limit using left/right hand limits

(i) Limit of functions involving absolute value




(ii) Limits of piecewise defined functions



(iii) Determining limits using a (or if given a) graph




5. Limits at Infinity ( determining end behaviour of functions , including sequences)



6. More Examples (only view 2:53 to 14:02)




7. Continuity


(i) Definition of Continuity




(ii) Some sample calculations involving continuity



(iii) Types of Discontinuity (lots of examples)
(Note: Her terminology is a little different from ours you can ignore 'essential' discontinuity)

1. Removable discontinuity = point discontinuity (or 'gap')
2. Infinite discontinuity = vertical asymptote




(iv) Another good example of a question involving continuity of a function




(v) A nice summary of functions we can ASSUME to be continuous (at each point in its domain)
(note: add polynomial, rational, absolute value and trigonometric functions to his list)

link to the video

7. Review of Limits and Continuity